Simple Pendulum undergoing harmonic oscillation

[Safder Aree]

Homework Statement

Is the time average of the tension in the string of the pendulum larger or smaller than
mg? By how much?

Homework Equations

$$F = -mgsin\theta $$
$$T = mgcos\theta $$

The Attempt at a Solution

I'm mostly confused by what it means by time average. However from my understanding of the question the answer should be yes it is smaller than mg. Since $$T=mgcos\theta$$.

Therefore, it is smaller by a factor of cos(theta) unless the angle is 0 or pi in that case it is equal to mg. Does that make sense?

 

[jambaugh]

I would suggest you consider the fact that the mass on the end of the pendulum does not have any net vertical acceleration across the period of one cycle.

 

[Safder Aree]

I would suggest you consider the fact that the mass on the end of the pendulum does not have any net vertical acceleration across the period of one cycle.

Sorry, I'm not sure I follow.

 

[jambaugh]

Picture the swinging pendulum. Picture the force due to tension acting on the bob, the mass m. That force as a vector changes direction a bit as the pendulum swings. As a vector it has a vertical component and horizontal component. Think about your question in terms of the time average of components, vertical and horizontal.

 

[haruspex]

That force as a vector changes direction

Tension is not a vector. However, I would not rule out the possibility that the question setter intended it to be taken as such.
@Safder Aree , there is something else to consider.
What are the forces on the mass when at the lowest point of the swing? What is its acceleration there?

 

[jambaugh]

Tension is not a vector.

which is why I didn't say it was. I said the force due to tension on the bob is a vector.
The tension is the magnitude of the named vector and the direction is defined by the angle of the pendulum as it swings.

@haruspex the force on the mass when at the lowest point of the swing will be a maximum but it will not give unambiguous information about the time average.

@Safder Aree Here's another hint. Imagine replacing the pendulum with an equal mass suspended from a spring with spring constant such that the mass spring system oscillates twice as fast as the pendulum... but if you look at their vertical components only, they match. The mass spring is modeling just the vertical component of the pendulum. The forces on the mass in the mass spring system will be just (about) the same as the vertical component of the force on the mass in the pendulum system.

(the frequency of the mass spring needs to be double because the pendulum reaches max height and min height twice per cycle.)

[EDIT: Actually this spring example may not be the easiest way there... consider this fact: Since the system is periodic, the time (vector) average of all forces on the mass must be zero over a period since its motion is periodic. All forces including the force of gravity. Think about what that means when you express the total force (not always zero but averaging to zero) as a sum of gravity's constant force and the string's (or spring's in my other example) force on the mass.

 

[haruspex]

which is why I didn't say it was. I said the force due to tension on the bob is a vector.

No, but you wrote this:

Think about your question in terms of the time average of components, vertical and horizontal.

Which is not going to work because tension is not a vector.

the force on the mass when at the lowest point of the swing will be a maximum but it will not give unambiguous information about the time average.

Consideration of that position should lead to the realisation that a contributor to tension has been omitted.

 

[ehild]

Homework Statement

Is the time average of the tension in the string of the pendulum larger or smaller than
mg? By how much?

Homework Equations

$$F = -mgsin\theta $$
$$T = mgcos\theta $$

No, the tension is not mgcos(θ),
The bob moves along a circle, so the net radial force should be equal to the centripetal force.

The Attempt at a Solution

I'm mostly confused by what it means by time average.

Time average is the integral to one period divided by the length of a period. For small diplacements, the angle of the string changes according to SHM: $θ=Asin(\frac{2π}{T}t+φ)$ . If the bob starts from the equilibrium position, φ=0.
You know how the the time period T of the bob is calculated from the length of the pendulum and the mass of the bob. The velocity of the bob is v=(dθ/dt) L. Once you get the time dependence of the velocity, you can write out the time dependence of the tension, and you can integrate it for one period.

 

[haruspex]

the angle of the string changes according to SHM:

you can integrate it for one period.

We are not told these are small oscillations, and the integral is quite nasty for the general case.
Fortunately we are not asked to find the average tension, merely how it compares with mg.

@jambaugh had the best idea - no integrals required - but I misunderstood the direction he was going because he mentioned averaging the horizontal component. The key is to average just the vertical component and consider how the existence of a horizontal component affects the average magnitude.

 

[Safder Aree]

We are not told these are small oscillations, and the integral is quite nasty for the general case.
Fortunately we are not asked to find the average tension, merely how it compares with mg.

@jambaugh had the best idea - no integrals required - but I misunderstood the direction he was going because he mentioned averaging the horizontal component. The key is to average just the vertical component and consider how the existence of a horizontal component affects the average magnitude.

I actually forgot to mention that we are to assume small oscillations. I'm still quite confused on where to proceed with this question. How would you go about averaging the vertical component.

Would T:
$$Tcos(\theta) = mg$$
$$Tsin(\theta) = F$$

Thus
$$T = \frac{mg}{cos(\theta)}$$

 

[haruspex]

I actually forgot to mention that we are to assume small oscillations.

Ok, but using @jambaugh's method you don't need to assume that.
Instead, think about vertical momentum.
If the average vertical force over a swing is F_y,2π, and the period is T_2π, what would be the net change in vertical momentum?
This approach is the easiest way of answering how the average tension compares with mg.

If you want to take the approach of calculating what the average tension is (assuming small oscillations):

Would T:
$$Tcos(\theta) = mg$$

No, that's not right either. Please don't make wild guesses.
You were closer with:

$$T = mgcos\theta $$

which is true at each end of the swing.
For a general instant of the swing, as I mentioned in post #5 and @ehild in post #8, you are overlooking centripetal force.
Draw a free body diagram of the general position. What is the net force in the radial direction? What is the acceleration in that direction?

 

[ehild]

We are not told these are small oscillations, and the integral is quite nasty for the general case.
Fortunately we are not asked to find the average tension, merely how it compares with mg.

The title of the thread is about "simple pendulum undergoing harmonic oscillation." And it is also asked: by how much is it smaller or greater than mg?

As for integration, my method uses only the fact that the integral of a periodic function is zero for a full period.
The problem can be solved also with conservation of energy, and then using small-angle approximation.
@ Safder Aree: But the first thing is to know how the tension depends on the position of the bob.

 

[haruspex]

it is also asked: by how much is it smaller or greater than mg?

You're right - I missed that. But then the question becomes, to which order of approximation?
I feel that being told it is SHM doesn't quite address that. Taking it as SHM, we might come up with an average tension that differs from mg in the θ₀² term. In that case, starting afresh, not taking it as SHM, but still applying a small angle approximation, we could get a different θ₀² term.

 

[ehild]

You're right - I missed that. But then the question becomes, to which order of approximation?
I feel that being told it is SHM doesn't quite address that. Taking it as SHM, we might come up with an average tension that differs from mg in the θ₀² term. In that case, starting afresh, not taking it as SHM, but still applying a small angle approximation, we could get a different θ₀² term.

I think a first order approximation would be quite satisfactory for the average tension.

Both methods should give the same second order term in θ₀.

 

[haruspex]

I think a first order approximation would be quite satisfactory for the average tension.

Even if that is just mg? Don't we need the first nonzero term beyond the zeroth order?

 

[ehild]

Even if that is just mg? Don't we need the first nonzero term beyond the zeroth order?

How do you know that the first nonzero term is quadratic in θ₀?

 

[haruspex]

How do you know that the first nonzero term is quadratic in θ₀?

That is the result I got.

 

[ehild]

That is the result I got.

You are right, it is quadratic in θ₀, but the same with both methods.

 

[haruspex]

You are right, it is quadratic in θ₀, but the same with both methods.

I get different results, but my two methods need not be the same as your two.

 

[ehild]

I get different results, but my two methods need not be the same as your two.

My first method was assuming SHM : θ=θ₀sin(ωt) with $ω=\sqrt{g/L}$
The other method was with conservation of energy.
I used the approximation sin(x)≈x, cos(x)≈1-x²/2.

 

[haruspex]

My first method was assuming SHM : θ=θ₀sin(ωt) with $ω=\sqrt{g/L}$
The other method was with conservation of energy.
I used the approximation sin(x)≈x, cos(x)≈1-x²/2.

Same here, but I can see other possible points of divergence.
No time right now, but I'll post my solutions in a private message, tomorrow probably.

 

[ehild]

Same here, but I can see other possible points of divergence.
No time right now, but I'll post my solutions in a private message, tomorrow probably.

I used time average.